Counting is now underway in two key Federal by-elections with the Liberal Party bracing for a possible voter backlash.

The polling booths have now closed in the safe blue-ribbon seats of Bradfield in Sydney and Higgins in Melbourne.

Campaigning this morning in Bradfield, new Liberal Leader Tony Abbott says he is confident of a win, but he expects a slight swing to the Greens.

But he says no-one should read anything into such a result because the Greens Party is the main opposition, with the ALP not standing candidates in either by-election.

Mr Abbott says the recent Liberal leadership turmoil is now resolved and the Coalition is united and has backed the new policy on climate change.

The blue-ribbon seat was held for over 10 years by former opposition leader Dr Brendan Nelson, who resigned earlier this year.

Despite predictions of a swing towards the Greens, Mr Fletcher says the people of Bradfield want a united Liberal Party that is able to take on the Government.

"I've heard that message very clearly, the Liberal Party has heard that message very clearly, we have a new leader in Tony Abbott," he said.

"Tony has my strong support and for us to be an effective and united party, the first thing that must happen is that every parliamentarian and every aspirant parliamentarian gets strongly behind Tony Abbott."

Greens play to strengths

While Mr Abbott only expects a slight swing against his party, Ms Gemmell says the Liberals have underestimated the importance of climate change to the people of Bradfield.

"I think Paul Fletcher's wrong to have campaigned saying the electorate doesn't care about climate change and I think he's wrong to say people are only looking at local issues," she said.

"There's a combination of issues that people are looking for, and primarily they're federal issues, because this is a federal campaign.

"Climate change has been the issue of the week and I think Paul Fletcher has missed the mark."

Result could be close

Dr Nelson said earlier today the party is not taking anything for granted in his old seat.

He said by-elections usually produce unexpected outcomes.

"We have a 6 per cent swing to us against the Labor Party in the Gippsland by-election last year - we had a swing against us in the by-election in Mayo. Anything could happen in Bradfield today," he said.

"But nothing has been left to chance by the Liberal Party, by Paul Fletcher who is the Liberal voice for Bradfield and we'll just see what result brings."

Former federal treasurer Peter Costello says Higgins will be won on local issues.

Liberal candidate Kelly O'Dwyer is expected to comfortably win the seat Mr Costello held for more than 20 years.

Canberra-based academic and author Clive Hamilton is contesting the seat for the Greens.

Mr Costello says the result could be close.

"This seat is marginal, it was marginal before tonight and it will be marginal after tonight," he said.

"I think this seat will be focusing on local issues. Sure there's been noise from Canberra, but my message to people is that this is still a fight for the seat of Higgins, for the suburbs of south-east Melbourne."